This volume is the latest edition in Palgrave Macmillan’s History of Analytic Philosophy series, and it deals exclusively with the philosophical thought of Anton Marty, a student of Franz Brentano at Würzburg and Hermann Lotze at Göttingen. The reason for such a volume is that Marty is often overlooked and underestimated. In both the analytic and phenomenological traditions, Brentano, Alexius Meinong and Edmund Husserl receive most of the attention and Marty is often seen as merely a defender of Brentano – not a philosopher in his own right. This book seeks to disrupt these preconceived notions about Marty, in a way that clearly demonstrates the promise of his ideas for contemporary research (for both the analytic and phenomenological traditions and beyond) while breathing “new life into his thought”. (vii) For example, pieces by François Recanati and Mark Textor highlight Marty’s original contributions while engaging in fresh critical discussion of his work alongside that of Paul Grice and Brentano. Kevin Mulligan does something similar with Ludwig Wittgenstein. Other authors, like Ingvar Johansson, showcase Marty’s contributions (for example, with space) that have been excluded from the history of philosophy. This volume feels less like a simple overview of a forgotten thinker and more like a critical introduction that simultaneously launches the reader into fruitful dialogue with both contemporary and longstanding issues in analytic philosophy. This book is organized into three parts: Issues pertaining to philosophy of language; philosophy of space and time; and the metaphilosophical aspects of existence and being in his thought.

In the first part, focusing on philosophy of language, Textor’s chapter stands out as particularly well executed, and which would appeal to a broader audience than just the analytic tradition. What is said here will be of great value to scholars in the phenomenological tradition who study the early work of Edmund Husserl or the Munich Circle students who studied with him before the outbreak of WWI. Issues surrounding the nature of language and signification, statements expressing wishing, commanding and questioning, and especially judgment are central to the works of Johannes Daubert and Adolf Reinach, who both read Marty, and then later students such as Roman Ingarden. Textor identifies Marty’s theory of language as ‘intentionalist semantics’ – Marty defined the word language as synonymous with intentional indication of the inner life of the person – and this metaphysical view of meaning comes with two commitments: first, mental facts concerning desire and belief are the most fundamental to what signs mean; and second, the speaker means something if and only if she does it with the purpose of producing an attitude for or in an audience. (34) This is where we see Marty and Grice roughly align. Textor focuses his essay on this second commitment – communicative intention –, but while he does so, he explores an alternative view of meaning put forth by Brentano. That is the idea that some utterances have meaning independently of whether they were made with the purpose of influencing others; therefore, with regard to the primary source of meaning, the utterance meaning takes priority over the speaker’s intended meaning for it. (35) Textor engages with Brentano’s position to remedy problems that both Marty and Grice fall prey to, specifically occurring with non-communicative utterances. Textor, however, isn’t painting Brentano as the answer to all of our problems, but rather delves into the shortcomings his view faces and then demonstrates how it can be rescued and developed to achieve greater insight about speaker meaning. He takes Brentano’s work on the meaning of utterances expressed in judgments and extends it, to create a model that will connect judgment and non-natural meaning, looking to the mechanism of belief acquisition. For example, if we believe a speaker to be trustworthy, we are more likely to make a rational judgment based on the information they share. Textor ends with: “There are further details to be filled in to complete Brentano’s picture, but I hope that I gave the reader some reasons to take Brentano’s proposal to be the basis for an alternative to Grice’s and Marty’s that is worth completing further.”(64) Textor primarily uses, as source material, Brentano’s logic lectures (EL 80), taken from the Würzburg course of the winter semester 1869/70 entitled Deduktive und Induktive Logik.[1] Brentano lectured for many years on logic, while at Würzburg and later Vienna, and it is great to see these lectures being highlighted and utilized. Here, we see their value communicated, and Textor provides his own (excellent) translations – this is more than simply a passing mention of Brentano’s academic teaching history.

This piece by Textor is a real gem, because the reader gets a thorough journey into theories of language that were happening just prior to the activities of Gottlob Frege and Edmund Husserl. For the latter, in particular, it was setting the stage for what he would write in the Logical Investigations (1900-1901). For an early phenomenology scholar like myself, this chapter is great for the discussion of Brentano logic lectures and the Marty writings that rarely receive any attention and yet have such a central role to play in the ideas of the early movement. Also, it is wonderful to read Brentano’s logical insights about language, and see them given serious consideration alongside someone like Grice, and in fact used to help Grice, as this work often takes a backseat to his intentionality thesis contained in the Psychology From an Empirical Standpoint and his later reism.

In the second part, the chapters on the consciousness of space and time are some of my favorite. This section was my reason for wanting a copy of this book, if I am honest. Once again, these chapters will appeal and prove very helpful to those in both the analytic and phenomenological traditions who wish to understand the discussions of the consciousness of time and space that informed major figures in the 20th century, and for me this means Husserl. This topic is yet another that Husserl lectured on early in the 20th century, and this theme continued to be a popular one with both the Munich and Göttingen Circles, for example in the works of Max Scheler, Moritz Geiger, Hedwig Conrad-Martius, and Roman Ingarden. It was also one that Henri Bergson wrote about, and it informed the position expressed during the famous debate with Einstein in Paris. In this second part of the volume the first essay by Johansson, “A Presentation and Defense of Anton Marty’s Conception of Space” goes beyond a defense of Marty. Johansson clearly demonstrates how Marty’s ideas on the topic are relevant and important not just to history of philosophy but also to the field of physics. As he points out at the beginning, there are two kinds of space: the perceptual and the physical. He focuses on the physical expression of space, bringing together ideas from Marty with elements of Immanuel Kant, Graham Nerlich, and himself to defend a container conception of space and space-time and to show why contemporary physics should give it serious consideration. Marty’s theory holds that space has a mind-independent existence, where all bodies, properties, events, spatial points and relations are all contained within this ontologically preexisting space. (100 – 102) He also leaves open the possibility that space could be empty, which goes against not only Kant but also Brentano. This position is opposed to the relational theory of space, which was held by Leibniz. While Marty’s theory most likely falls under the modern label “substantivalism” (i.e., the theory that space exists in itself in addition to the material objects within it), it doesn’t fit squarely: while it can conform to the general definition of substantivalism, Marty’s conception of space is ontologically more basic or, rather, primary to what is contained within it, and this makes him distinct from other “substantivalists”, like Barry Dainton. By the way, there is a great discussion of Dainton here, too. This chapter offers a wonderful historical run down, along with comparison of Marty’s conception, and in such an accessible way. If you are rusty on the topic or new to it, this chapter is a great primer and will also leave you with some points to think about.

The next essay by Clare Mac Cumhaill “Raum and ‘Room’: Comments on Anton Marty on Space Perception” is the perfect follow up to Johansson. Cumhaill’s piece elaborates and extends what Johansson discussed, in particular on perception, and then in the comparisons of Marty to others who write on space and time, and again in a very approachable and engaging way. The essay contains an informative outline of Marty’s conception of the ontology of space, a section on Marty’s critiques of Kant and Brentano on the topic of space and time, and an inquiry into whether any contemporary theory of perception can handle Marty’s notion of space and time. The most promising for Cumhaill is Naïve Realism, but this comes with its own difficulties. A highlight for me was the section comparing Husserl and Marty; it was full of insights. I actually wanted more Husserl and comparison talk of him, because of what I stated earlier, but what is there is great (in particular on 137, the sections of the letters Marty wrote to Husserl are a fun read).

Thomas Sattig closes out this part of the volume with a bang, with his chapter: “Experiencing Change: Extensionalism, Retentionalism, and Marty’s Hybrid Account.” Sattig builds on the previous two chapters to discussing contemporary ideas concerning our experience of change: after some helpful encapsulations of extentionalism and retentionalism, there is a wonderful summary of Marty’s account, and at the close there are some challenges raised against Marty’s view. Marty’s position is called a “hybrid account” because, as pointed out in section three, the notion of how we experience change combines elements from both the extensionalist and retentionalist views, and in a presentist framework (i.e., only the present is actual, the past and future are not). (163) This chapter, like the others, is well organized, accessible and has an engaging style; it even has some lovely diagrams with leaves to help illustrate (great diagrams are necessary for discussions of time). The challenges to Marty’s view are excellent, and the suggested fixes for the holes or omissions in Marty’s theory offered are thorough, but Sattig also leaves room for the reader to think and form their own insights about these shortcomings.

While I only discussed chapters from the first two sections of this book, this should not in anyway convey to anyone reading this review that the third section is subpar or weak – it isn’t. The reader will get more fantastic pieces that really turn the spotlight on Marty’s work, which is much needed and deserved.

I really enjoyed what this volume had to offer and it reminded me of why I found Marty invaluable and fascinating during my graduate and postgraduate work. He’s an amazing talent and brilliant scholar in his own right, not simply a defender of Brentano and fellow priest who left the cloth with convictions about the infallibility of the pope. I really appreciated how this book was organized, and enjoyed how the chapters in each section relate but thoughtfully expand in various directions. The discussion of Marty is always balanced; the presentation of Marty feels very well rounded, and the contributors are always willing to talk about the errors as much as the successes. Furthermore, the fact that much of this book contains his lesser-known works is fantastic and asset to any collection or library. This volume also offers some great excursions into the history of philosophy, and this not only provides the context for Marty’s ideas but also what made him such a great philosopher.

If I have anything critical to say, besides wanting more Husserl, it is that some might come to the idea that Marty is an analytic philosopher or more of a forefather to the analytic tradition than to phenomenology or any other discipline. This can be gathered by the title of the book series and then the index of authors cited in the chapters. The introduction to this volume tries to convey that this is not what is being argued; it attempts to show that Marty’s work had significant influence on the analytic tradition, more influence than we currently feel he had, given that so much of his work is overlooked. But once you get into chapters, it is easy to forget what was said in the introduction and jump to conclusions, because sometimes the feel or approach is itself very analytic. However, I will say, it would be shortsighted to jump to such conclusions and/or to not to read this book. This volume offers a wonderful picture of Marty that is insightful, thought provoking, and inspirational. As I said many times (proportionally to how many times I noticed this in my reading), it is also an approachable and engaging to read. As a scholar of Husserl and Reinach, I see a lot of potential ties to my own work. Marty is one of many forefathers that both the analytic and phenomenological traditions share, and we should celebrate this man and his mind rather than divide ourselves into camps. Hey, we both share great taste in Austrians of the 19th century! Brentano and his students were immensely productive, interdisciplinary and incredibly brilliant; they changed the 20th century dialogue for philosophy – period. That being said, I highly recommend this book for both scholars of analytic philosophy and phenomenology, as well as those interested in the topics discussed between its covers.

This is a book long overdue. Other authors have made more or less recent phenomenological and transcendental-idealist contributions to the philosophy of mathematics: Dieter Lohmar (1989), Richard Tieszen (2005) and Mark van Atten (2007) are perhaps the most important ones. Ten years is a sufficiently wide gap to welcome any new work. Yet da Silva’s contribution stands out for one reason: it is unique in the emphasis it puts, not so much, or not only, on the traditional problems of the philosophy of mathematics (ontological status of mathematical objects, mathematical knowledge, and so on), but on the problem of the application of mathematics. The author’s chief aim – all the other issues dealt with in the book are subordinated to it – is to give a transcendental phenomenological and idealist solution to the evergreen problem of how it is that we can apply mathematics to the world and actually get things right – particularly mathematics developed in complete isolation from mundane, scientific or technological efforts.

Chapter 1 is an introduction. In Chapters 2 and 3, da Silva sets up his tools. Chapters 4 to 6 are about particular aspects of mathematics: numbers, sets and space. The bulk of the overall case is then developed in Chapters 7 and 8. Chapter 9, “Final Conclusions”, is in fact a critique of positions common in the analytic philosophy of mathematics.

Chapter 2, “Phenomenology”, is where da Silva prepares the notions he will then deploy throughout the book. Concepts like intentionality, intuition, empty intending, transcendental (as opposed to psychological) ego, and so on, are presented. They are all familiar from the phenomenological literature, but da Silva does a good job explaining their motivation and highlighting their interconnections. The occasional (or perhaps not so occasional) polemic access may be excused. The reader expecting arguments for views or distinctions, however, will be disappointed: da Silva borrows liberally from Husserl, carefully distinguishing his own positions from the orthodoxy but stating, rather than defending, them. This creates the impression that, at least to an extent, he is preaching to the converted. As a result, if you are looking for reasons to endorse idealism, or to steer clear of it, this may not be the book for you.

Be that as it may, the main result of the chapter is, unsurprisingly, transcendental idealism. This is the claim that, barring the metaphysical presuppositions unwelcome to the phenomenologist, there is nothing more to the reality of objects than their being “objective”, i.e., public. ‘Objectivation’, as da Silva puts it, ‘is an intentional experience performed by a community of egos operating cooperatively as intentional subjects. … Presentifying to oneself the number 2 as an objective entity is presentifying it and simultaneously conceiving it as a possible object of intentional experience to alter egos (the whole community of intentional egos)’ (26-27). This is true of ideal objects, as in the author’s example, but also of physical objects (the primary type of intentional experience will then be perception).

There are two other important views stated and espoused in the chapter. One is the Husserlian idea that a necessary condition for objective existence is the lack of cancellation, due to intentional conflict, of the relevant object. Given the subject matter of the book, the most important corollary of this idea is that ideal objects, if they are to be objective, at the very least must not give rise to inconsistencies. For example, the set of all ordinals does not objectively exist, because it gives rise to the Burali-Forti paradox. The other view, paramount to the overall case of the book (I will return to it later), is that for a language to be material (or materially determined) is for its non-logical constants to denote materially determined entities (59). If a language is not material, it is formal.

Chapter 3 is about logic. Da Silva attempts a transcendental clarification of what he views as the trademark principles of classical logic: identity, contradiction and bivalence. The most relevant to the book is the third, and the problem with it is: how can we hold bivalence – for every sentence p, either p or not-p – and a phenomenological-idealist outlook on reality? For bivalence seems to require a world that is, as da Silva puts it, ‘objectively complete’: such that any well-formed sentence is in principle verifiable against it. Yet how can the idealist’s world be objectively complete? Surely if a sentence is about a state of affairs we currently have no epistemic access to (e.g., the continuous being immediately after the discrete) there just is no fact of the matter as to whether the sentence is true or false: for there is nothing beyond what we, as transcendental intersubjectivity, have epistemic access to.

Da Silva’s first move is to put the following condition on the meaningfulness of sentences: a sentence is meaningful if and only if it represents a possible fact (75). The question, then, becomes whether possible facts can always be checked against the sentences representing them, at least in principle. The answer, for da Silva, turns on the idea, familiar from Husserl, that intentional performances constitute not merely objects, but objects with meanings. This is also true of more structured objectivities, such as states of affairs and complexes thereof – a point da Silva makes in Chapter 2. The world (reality) is such a complex: it is ‘a maximally consistent domain of facts’ (81). The world, then, is intentionally posited (by transcendental intersubjectivity) with a meaning. To hold bivalence as a logical principle means, transcendentally, to include ‘objective completeness’ in the intentional meaning (posited by the community of transcendental egos) of the world. In other words, to believe that sentences have a truth value independent of our epistemic access to the state of affairs they represent is to believe that every possible state of affairs is in principle verifiable, in intuition or in non-intuitive forms of intentionality. This, of course, does not justify the logical principle: it merely gives it a transcendental sense. Yet this is exactly what da Silva is interested in, and all he thinks we can do. Once we refuse to assume the objective completeness of the world in a metaphysical sense, what we do is to assume it as a ‘transcendental presupposition’ or ‘hypothesis’. In the author’s words:

How can we be sure that any proposition can be confronted with the facts without endorsing metaphysical presuppositions about reality and our power to access reality in intuitive experiences? … By a transcendental hypothesis. By respecting the rules of syntactic and semantic meaning, the ego determines completely a priori the scope of the domain of possible situations – precisely those expressed by meaningful propositions – which are, then, hypothesized to be ideally verifiable. (83)

Logical principles express transcendental hypotheses; transcendental hypotheses spell out intentional meaning. … The a priori justification of logical principles depends on which experiences are meant to be possible in principle, which depends on how the domain of experience is intentionally meant to be. (73)

There is, I believe, a worry regarding da Silva’s definition of meaningfulness in terms of possible situations: it seems to be in tension with the apparent inability of modality to capture fine-grained (or hyper-) intensional distinction and therefore, ultimately, meaning (for a non-comprehensive overview of the field of intensional semantics, see Fox and Lappin 2005).[1] True, since possible situations are invoked to define the meaningfulness, not the meaning, of sentences, there is no overt incompatibility; yet it would be odd to define meaningfulness in terms of possible situations, and meaning in a completely different way.

Chapter 4, “Numbers”, has two strands. The first deals with another evergreen of philosophy: the ontological status of numbers and mathematical objects in general. Da Silva’s treatment is interesting and his results, as far as I can see, entirely Husserlian: numbers and other mathematical objects behave like platonist entities except that they do not exist independently of the intentional performances that constitute them. One consequence is that mathematical objects have a transcendental history which can and should be unearthed to fully understand their nature. The phenomenological approach is unique in its attention to this interplay between history and intentional constitution, and it is to da Silva’s credit, I believe, that it should figure so prominently in the book. Ian Hacking was right when he wrote, a few years back, that ‘probably phenomenology has offered more than analytic philosophy’ to understand ‘how mathematics became possible for a species like ours in a world like this one’ (Hacking 2014). Da Silva’s work fits the pattern.

And yet I have a few reservations, at least about the treatment (I will leave the results to readers). For one thing, there is no mention of unorthodox items such as choice sequences. Given da Silva’s rejection of intuitionism in Chapter 3, perhaps this is unsurprising. Yet not endorsing is one thing, not even mentioning is quite another. I cannot help but think the author missed an opportunity to contribute to one of the most engaging debates in the phenomenology of mathematics of the last decade (van Atten’s Brouwer Meets Husserl is from 2007). Da Silva’s seemingly difficult relationship with intuitionism is also connected with another conspicuous absence from the book. At p. 118 da Silva looks into the relations between our intuition of the continuum and its mathematical construction in terms of ‘tightly packed punctual moments’, and argues that the former does not support the latter (which should then be motivated on different grounds). He cites Weyl as the main purveyor of an alternative model – which he might well be. But complete silence about intuitionist analysis seems frankly excessive.

A final problem with da Silva’s presentation is his dismissal of logicism as a philosophy of, and a foundational approach to, mathematics. ‘Of course,’ he writes, ‘Frege’s project of providing arithmetic with logical foundations collapsed completely in face of logical contradiction (Russell’s paradox)’ (103). The point is not merely historical: ‘Frege’s reduction of numbers to classes of equinumerous concepts is an unnecessary artifice devised exclusively to satisfy logicist parti-pris … That this caused the doom of his projects indicates the error of the choice’. I would have expected at least some mention of either Russell’s own brand of logicism (designed, with type theory, to overcome the paradox), or more recent revivals, such as Bob Hale’s and Crispin Wright’s Neo-Fregeanism (starting with Wright 1983) or George Bealer’s less Fregean work in Quality and Concept (1982). None of these has suffered the car crash Frege’s original programme did, and all of them are still, at least in principle, on the market. True, da Silva attacks logicism on other grounds, too, and may argue that, in those respects, the new brands are just as vulnerable as the old. Yet, that is not what he does; he just does not say anything.

The second strand of the chapter, more relevant to the overall case of the book, develops the idea that numbers may be regarded in two ways: materially and formally. The two lines of investigation are not totally unrelated, and indeed some of da Silva’s arguments for the latter claim are historical. The claim itself is as follow. According to da Silva, numbers are essentially related to quantity: ‘A number is the ideal form that each member of a class of equinumerous quantitative forms indifferently instantiates’, and ‘two numbers are the same if they are instantiable as equinumerical quantitative forms’ (104).[2] Yet some types of numbers are more or less detached from quantity: if in the case of the negative integers, for example, the link with quantity is thin, when it comes to the complex numbers it is gone altogether. Complex numbers are numbers only in the sense that they behave operationally like ones – but they are not the real (no pun intended) thing. Da Silva is completely right in saying that it was this problem that moved the focus of Husserl’s reflections in the 1890s from arithmetic to general problems of semiotic, logic and knowledge. The way he cashes out the distinction is in terms of a material and a formal way to consider numbers. Genuine, ‘quantitative’ numbers are material numbers. Numbers in a wider sense, and thus including the negative and the complex, are numbers in a formal sense. Since, typically, the mathematician is interested in numbers either to calculate or because they want to study their relations (with one another or with something else), they will view numbers formally – i.e., at bottom, from the point of view of operations and structure – rather than materially.

Thus, the main theoretical result of the chapter is that, inasmuch as mathematics is concerned with numbers, it is ‘essentially a formal science’ (120). In Chapter 7, da Silva will put forward an argument to the effect that mathematics as a whole is essentially a formal science. This, together with the idea, also anticipated in Chapter 4, that the formal nature of mathematics ‘explains its methodological flexibility and wide applicability’, is the core insight of the whole book. But more about it later.

Chapter 5 is about sets. In particular, da Silva wants to transcendentally justify the ZFC axioms. This includes a (somewhat hurried) genealogy, roughly in the style of Experience and Judgement, of ‘mathematical sets’ from empirical collections and ‘empirical sets’. The intentional operations involved are collecting and several levels of formalisation. The details of the account have no discernible bearing on the overarching argument, so I will leave them to one side. It all hinges, however, on the idea that sets are constituted by the transcendental subject through the collecting operation, and this is what does the main work in the justification. This makes da Silva’s view very close to the iterative conception (as presented for example in Boolos 1971); yet he only mentions it once and in passing (146). Be that as it may, it is an interesting feature of da Silva’s story that it turns controversial axioms such as Choice into sugar, while tame ones such as Empty Set and Extensionality become contentious.

Empty Set, for example, is justified with an account, which da Silva attributes to Husserl, of the constitution of empty sets that I found fascinating but incomplete. Empty sets are clearly a hard case for the phenomenological account: because, as one might say, since collections are empty by definition, no collecting is in fact involved. Or is it? Consider, da Silva says, the collection of the proper divisors of 17:

Any attempt at actually collecting [them] ends up in collecting nothing, the collecting-intention is frustrated. Now, … Husserl sees the frustration in collecting the divisors of 17 as the intuitive presentation of the empty collection of the divisors of 17. So empty collections exist. (148)

It is a further question, and da Silva does not consider it, whether this story accounts for the uniqueness of the empty set (assuming he thinks the empty set is indeed unique, which, as will appear, is not obvious to me). Are collecting-frustration experiences all equal? Or is there a frustration experience for the divisors of 17, one for the divisors of 23, one for the round squares, and so on? If they are all equal, does that warrant the conclusion that the empty sets they constitute are in fact identical? If they are different, what warrants that conclusion? Of course, an option would be: it follows from Extensionality. Yet, I venture, that solution would let the phenomenologist down somewhat. More seriously, da Silva even seems to reject Extensionality (and thus perhaps the notion that there is just one empty set). At least: he claims that there is ‘no a priori reason for preferring’ an extensional to an intensional approach to set theory, but that if we take ‘the ego and its set-constituting experiences’ seriously we ought to be intensionalists (150).

Chapters 6 is about space and its mathematical representations – ‘a paradigmatic case of the relation between mathematics and empirical reality’ (181). It is where da Silva deals the most with perception and the way it relates with mathematical objects. For the idealist, there are at least four sorts of space: perceptual, physical, mathematical-physical and purely formal. The intentional action required to constitute them is increasingly complex, objectivising, idealising and formalising. Perceptual space is subjective, i.e., private as opposed to public. It is also ‘continuous, non-homogeneous, simply connected, tridimensional, unbounded and approximately Euclidean’ (163). Physical space is the result of the intersubjective constitution of a shared spatial framework by harmonization of subjective spatial experiences. This constitution is a ‘non-verbal, mostly tacit compromise among cooperating egos implicit in common practices’ (167). Unlike its perceptual counterpart, physical space has no centre. It also admits of metric, rather than merely proto-metric, relations. It is also ‘everywhere locally’, but not globally, Euclidean (168). The reason is that physical space is public, measurable but based merely on experience (and more or less crude methods of measurement) – not on models.

We start to see models of physical space when we get to mathematical-physical space. In the spirit of Husserl’s Krisis, da Silva is very keen on pointing out that mathematical-physical space, although it does indeed represent physical space, does not reveal what physical space really is. That it should do so, is a naturalistic misunderstanding. In the author’s words:

At best, physical space is proto-mathematical and can only become properly mathematical by idealization, i.e., an intentional process of exactification. However, and this is an important remark, idealization is not a way of uncovering the “true” mathematical skeleton of physical space, which is not at its inner core mathematical. (169)

Mathematical-physical space is what is left of the space we live in – the space of the Lebenswelt, if you will – in a representation designed to make it exact (for theoretical or practical purposes). Importantly, physical space ‘sub-determines’ mathematical-physical space: the latter is richer than the former, and to some extent falsifies what it seeks to represent. Euclidean geometry is paradigmatic:

The Euclidean representation of physical space, despite its intuitive foundations, is an ideal construct. It falsifies to non-negligible extent perceptual features of physical space and often attributes to it features that are not perceptually discernible. (178)

The next step is purely formal representations of space. These begin by representing physical space, but soon focus on its formal features alone. We are then able to do analytic geometry, for example, and claim that, ‘mathematically, nothing is lost’ (180). This connects with da Silva’s view that mathematics is a formal science and, in a way, provides both evidence for and a privileged example of it. If you are prepared to agree that doing geometry synthetically or analytically is, at bottom, the same thing, then you are committed to explain why that is so. And da Silva’s story is, I believe, a plausible candidate.

Chapter 7 is where it all happens. First, and crucially, da Silva defends the view that mathematics is formal rather than material in character. I should mention straight away that his argument, a three-liner, is somewhat underdeveloped. Yet it is very clear. To say that mathematics is essentially formal is, for da Silva, to say that mathematics can only capture the formal aspects of reality (as the treatment of space is meant to show). The reason is as follows. Theories are made up of symbols, which can be logical or non-logical. The non-logical symbols may, in principle, be variously interpreted. A theory whose non-logical symbols are interpreted is, recall, material rather than formal. Therefore, one could argue, number theory should count as material. Yet, so da Silva’s reasoning goes, ‘fixing the reference of the terms of an interpreted theory is not a task for the theory itself’ (186). The theory, in other words, cannot capture the interpretation of its non-logical constant: that is a meta-theoretical operation. But then mathematical theories cannot capture the nature, the specificity of its objects even when these are material.

That is the master argument, as well as the crux of the whole book. For it follows from it that mathematics is essentially about structure: objects in general and relations in which they stand. This, for da Silva, does not mean that mathematics is simply not about material objects. That would be implausible. Rather, the claim is that even when a mathematical theory is interpreted, or has a privileged interpretation, and is therefore about a specific (‘materially filled’) structure, it does not itself capture the interpretation (the fixing of it) – and thus it is really formal. Some mathematical theories are, however, formal in a stricter sense: they are concerned with structures that are kept uninterpreted. These are purely formal structures. Regarding space, Hilbert’s geometry is a good example.

Da Silva’s solution to the problem of the applicability of mathematics is thus the following. Mathematics is an intentional construction capable of representing the formal aspects of other intentional constructions – mathematics itself and reality. Moreover, it is capable of representing only the formal aspects of mathematics and reality. It should then be no surprise, much less a problem, that any non-mathematical domain can be represented mathematically: every domain, insofar as it is an intentional construction, has formal aspects – which are the only ones that count from an operational and structural standpoint.

This has implications for the philosophy of mathematics. On the ground of his main result, da Silva defends a phenomenological-idealist sort of structuralism, according to which structures are the privileged objects of mathematics. Yet his structuralism is neither in re nor ante rem. Not in re, because structures, even when formal, are objects in their own right. Not ante rem, because structures are intentional constructs, and thus not ontologically independent. They depend on intentionality, but also on the material structures on whose basis they are constituted through formalisation. This middle-ground stance is typical of phenomenology and transcendental idealism.

I have already said what the last two chapters – 8 and 9 – are about. The latter is a collection of exchanges with views in the analytic philosophy of mathematics. They do not contribute to the general case of the book, so I leave them to prospective readers. The former is an extension of the results of Chapter 7 to science in general. A couple of remarks will be enough here. Indeed, when the reader gets to the chapter, all bets are off: by then, da Silva has put in place everything he needs, and the feeling is that Chapter 8, while required, is after all mere execution. This is not to understate da Silva’s work. It is a consequence of his claim (217) that the problem of the applicability of mathematics to objective reality, resulting in science, just is, at bottom, the problem of the applicability of mathematics to itself – which the author has already treated in Chapter 7. Under transcendental idealism, objective, physical reality, just like mathematical reality, is an intersubjective intentional construct. This construct, being structured, and thus having formal aspects to it, ‘is already proto-mathematical’ and, ‘by being mathematically represented, becomes fully mathematical’ (226). The story is essentially the same.

Yet it is only fair to mention that, while in this connection it would have been easy merely to repeat Husserl (the approach is after all pure Krisis), that is not what da Silva does. He rather distances himself from Husserl in at least two respects. First of all, he rejects what we may call the primacy of intuition in Husserl’s epistemology of mathematics and science. Second, he devotes quite a bit of space to the heuristic role of mathematics in science – made possible, so the author argues, by the formal nature of mathematical representation (234).

As a final remark, I want to stress again what seems to me the chief problem of the book. Da Silva’s aim is to give a transcendental-idealist solution to the problem of the applicability of mathematics. Throughout the chapters, he does a good job spelling out the details of the project. Yet there is no extensive discussion of why one should endorse transcendental idealism in the first place. True, a claim the author repeatedly makes is that idealism is the only approach that does not turn the problem into a quagmire. While the reader may be sympathetic with that view (as I am), da Silva offers no full-blown argument for it. As a result, the book is unlikely to build bridges between phenomenologists and philosophers of mathematics of a more analytic stripe. Perhaps that was never one of da Silva’s aims. Still, I believe, it is something of a shame.

[1] Unless impossible worlds are brought in – but as far as I can see that option is foreign to da Silva’s outlook.

[2] The notion of quantitative form is at the heart of Husserl’s own account of numbers in Philosophy of Arithmetic – and it is to da Silva’s credit that he takes Husserl’s old work seriously and accommodates into an up-to-date phenomenological-idealist framework.